Proof of the Yang-Mills Existence and Mass Gap Conjecture via Self-Adjoint Operator Spectral Theory

We prove the full Yang-Mills existence and mass gap conjecture in four-dimensionalMinkowski spacetime. Extending the self-adjoint operator spectral theory framework developed forthe Riemann Hypothesis and the Birch-Swinnerton-Dyer Conjecture, we construct a sequence offinite-dimensional self-adjoint matrices from the path integral of the Yang-Mills field. We establisha strict spectral correspondence between the eigenvalues of these matrices and the energy levelsof the Yang-Mills system. Using mathematical induction, the monotone convergence theorem forself-adjoint operators, and a novel energy regularization method, we extend these results to theinfinite-dimensional case, proving the existence of a global, smooth solution to the Yang-Millsequations and the existence of a non-zero mass gap in the energy spectrum. This result provides arigorous mathematical foundation for the standard model of particle physics. 2020 Mathematics Subject Classification. 81T13; 47B25; 47A10; 58J50. Key words and phrases. Yang-Mills theory; mass gap; self-adjoint operators; spectral decomposition; Millennium Prize Problems.